Antonio Zucco (originally just "Boss Zucco" in his first appearance) is a Mafia boss, or simple low level thug (his position of power varies depending upon the Golden and Silver Age continuity) in Gotham City who is responsible for the death of Dick Grayson's parents.
Throughout the years, Zucco's role in Robin's origin remains largely the same. Zucco tries to extort the Haley's Circus, where the Flying Graysons are the main attraction. When the ringleader, Mr. Haley, refuses to pay him protection money, Zucco sabotages the ropes the Graysons use in their act. They break while Dick's parents are in mid-air, sending them falling to their deaths.
Dick is subsequently adopted by Bruce Wayne (a.k.a. Batman) and becomes his partner Robin. Both of them locate Tony Zucco and send him to jail. In the Golden Age canon, he is sent to the electric chair, whereas he is sent to prison for life in the Silver Age version.
Batman: Dark Victory
In this series, Zucco (dubbed Anthony "Fats" Zucco) is portrayed as a low-level thug working for Sal Maroni, a member of Carmine Falcone's Mafia empire. He and another minor family head, Edward Skeevers, are put in charge of drug trafficking. After constant attacks by other enemies such as the Penguin, Zucco starts a new method of smuggling these drugs. He only lets Skeevers in on the secret, believing that its success will make their minor families greater than the Falcones and the Maronis put together, and attempts to take over Haley's Circus in order to use its trucks for drug trafficking. He is shown to be heavyset (hence his street name) and quite nervous, as he is constantly sweating. He kills Dick Grayson's parents in order to show the circus owner that he means business, and then quickly goes underground. He is later found by Batman and Dick Grayson, who is not yet Robin, and is chased down a dark alley by Grayson until he has a heart attack, confessing to various crimes.
Batman: Year Three
Although Dark Victory was meant to update Zucco's fate in Batman: Year Three, the Chuck Dixon miniseries Robin: Year One revealed Zucco did not die of a heart attack, but he actually went into a coma and was later sent to prison after recovering . He is sentenced to two consecuative life terms for the deaths of John and Mary Grayson, and he is incarcerated in Blackgate Penetientary.
Seven years after killing the Graysons (it was said he had served slightly over seven years, since Alfred says "It was almost ten years ago that Richard Grayson became a victim"), Zucco files for a parole hearing, offering testimony to put other criminals away. He professes remorse for his crimes, but his real motivations are for his own personal gain. Before he was sent to prison, Zucco had hidden a ledger detailing every operation in Gotham City in the orphanage Dick spent a month in after his parents' deaths. It would be enough to send any criminal boss to life imprisonment if he offered it to the authorities. The orphanage is being condemned and he is eager to get it back before the place is torn down. Wanting to keep the truth from Dick, who has recently become Nightwing, Alfred goes to the parole hearing and pleads for the judge to keep Zucco behind bars. Unfortunately, Dick does learn of Zucco being released early, despite Alfred's attempts, and he races to Blackgate to confront his parents' murderer once and for all. However, as he steps out of the prison, Zucco is gunned down by a helicopter hired by a rival crime boss to eliminate him. While it does give Dick some closure, his relationship with Batman is even further strained, even though Bruce insists he knew nothing about the parole.